A Modular Random-Key Framework for the Two-Dimensional Variable Sized Bin Packing Problem

We propose a modular metaheuristic framework based on random-key encoding to solve both the one- and two-dimensional Variable-Sized Bin Packing Problem (VSBPP). The VSBPP generalizes the classical bin packing problem by allowing bins of different sizes and associated costs, with the objective of minimizing total packing cost. Our method separates the optimization engine from problem-specific constraints via dedicated decoders and incorporates the No-Fit Polygon and Bottom-Left placement rules to address geometric feasibility and item rotation cuts in the 2D case. We benchmark the approach against state-of-the-art algorithms using four standard datasets. The proposed framework achieved average cost gaps between 0.15% and 0.40% for 1D instances and between 9.4% and 19.4% for 2D instances—closely approaching the performance of exact methods while using only 600 seconds of runtime. These results confirm the framework’s flexibility,
computational efficiency, and applicability to complex real-world packing problems.